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Unformatted text preview: “TAKE HOME” PORTION OF THE FINAL EXAM, DUE ON EXAM DAY (1) Let Rd be the (infinite) poset, whose ground set is the set of d-tuples of real numbers, and (a1 , . . . , ad ) ≤ (b1 , . . . , bd ) if and only if ai ≤ bi for all i = 1, . . . , d. Prove that the dimension of a poset P is equal to the least d such that P can be embedded1 into Rd . (2) Let X = {a, b, c, d, e, f } be the ground set of a poset. Define the (strict, i.e. irreflexive) relation to be {(a, b), (a, c), (a, d), (b, d), (e, c), (e, d), (e, f ), (f, d)}. Show that the resulting poset is of dimension 3. (3) Show that for n ≥ 3, if you remove a point from Sn , the resulting poset is of dimension n − 1. 1An embedding of a poset A into a poset B is an injective order-preserving map f : A → B . Order-preserving means that x < y in A if and only if f (x) < f (y ) in B . 1 ...
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This note was uploaded on 01/12/2012 for the course MATH 681 taught by Professor Wildstrom during the Fall '09 term at University of Louisville.

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